TITLE: Is it true that $\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap \mathbb{Q}$?
QUESTION [5 upvotes]: I know $\mathbb{Z}_{(p)}\subset \mathbb{Z}_{p}\cap \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at prime ideal $(p)$ and $\mathbb{Z}_p$ is the set of p-adic integers. I wonder if the converse inclusion holds. I prefer an explanation where $\mathbb{Z}_p$ is thought as the inverse limit. Thanks!

REPLY [4 votes]: Every subring of $\mathbb{Q}$ is a localization of $\mathbb{Z}$, namely at those prime numbers which become invertible. Here $p$ is the only exception. QED